OFFSET
0,2
COMMENTS
Related to Incomplete Gamma Function at 1/2. - Michael Somos, Mar 26 1999
For positive n, a(n) is equal to 2^n times the permanent of the n X n matrix with 3/2's along the main diagonal, and 1's everywhere else. - John M. Campbell, Jul 09 2011
Number of ways to sort a spreadsheet with n columns. (A subset of columns is chosen to sort on. These columns are ordered from major to minor, and each designated as to whether to sort by ascending or descending order. For example a spreadsheet with columns A,B,C,D could be sorted by column D ascending, then by column B descending, or any of 632 other ways.) - Marc LeBrun, Dec 07 2013
a(n) is a specific instance of sequences having the form b(0) = x, b(n) = a*n*b(n-1) + k for n >= 1. (Here x = 1, a = 2, and k = 1). Sequences of this form have a closed form of b(n) = n!*a^n*x + k*Sum_{j=1..n} n!*a^(n-j)/j!. - Gary Detlefs, Mar 26 2018
REFERENCES
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 262.
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..400
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 262.
Roland Bacher, Counting Packings of Generic Subsets in Finite Groups, Electr. J. Combinatorics, 19 (2012), #P7. - From N. J. A. Sloane, Feb 06 2013
Paul Barry, Eulerian-Dowling Polynomials as Moments, Using Riordan Arrays, arXiv:1702.04007 [math.CO], 2017.
Mathieu Guay-Paquet and Jeffrey Shallit, Avoiding Squares and Overlaps Over the Natural Numbers, Discrete Math., 309 (2009), 6245-6254. [From N. J. A. Sloane, Nov 27 2009]
Mathieu Guay-Paquet and Jeffrey Shallit, Avoiding Squares and Overlaps Over the Natural Numbers, arXiv:0901.1397 [math.CO], 2009.
Guo-Niu Han, Enumeration of Standard Puzzles [Wayback Machine link added by Felix Fröhlich, Mar 26 2018]
Guo-Niu Han, Enumeration of Standard Puzzles. [Cached copy]
Guo-Niu Han, Enumeration of Standard Puzzles, arXiv:2006.14070 [math.CO], 2020.
Michael Z. Spivey and Laura L. Steil, The k-Binomial Transforms and the Hankel Transform, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.1.
FORMULA
a(n) = floor(n! * e^(1/2) * 2^n) = n! * Sum_{k=0..n} 2^(n-k) / k! (i.e. binomial transform of (2n)!! = n!*2^n) = n! * (e^(1/2) * 2^n - Sum_{k >= n+1} 2^(n-k) / k!)). - Michael Somos, Mar 26 1999
E.g.f.: exp(x)/(1 - 2*x). - Vladeta Jovovic, Aug 11 2002
Sum_{n >= 1} 1/a(n) = 0.4246665348160769533082551230... - Cino Hilliard, Aug 19 2003
a(n) = Sum_{k=0..n} P(n, k)*2^k, where P(n,k) = n!/(n-k)!. - Ross La Haye, Aug 29 2005
G.f.: 1/(1 - x - 2*x/(1 - 2*x/(1 - x - 4*x/(1 - 4*x/(1 - x - 6*x/(1 - 6*x/(1 - x - 8*x/(1 - 8*x/(1 - x - 10*x/(1 - ... (continued fraction).
G.f.: 1/Q(0), where Q(k) = 1 - x*(4*k+3) - 4*x^2*(k+1)^2/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Sep 30 2013
a(n) = Sum_{k=0..n} C(n,k)*k!*2^k. - Marc LeBrun, Dec 07 2013
0 = a(n)*(2*a(n+1) - 5*a(n+2) + a(n+3)) + a(n+1)*(a(n+1) + a(n+2) - a(n+3)) + a(n+2)*a(n+2) if n > -2. - Michael Somos, Jan 02 2014
a(n) + (-2*n-1)*a(n-1) + 2*(n-1)*a(n-2) = 0. - R. J. Mathar, Jan 31 2014
a(n) = hypergeometric_U(1, n+2, 1/2)/2. - Peter Luschny, Nov 26 2014
From Peter Bala, Jan 30 2015: (Start)
a(n) = Integral_{x >= 0} (2*x + 1)^n*exp(-x) dx. (Cf. A000354.)
The e.g.f. y = exp(x)/(1 - 2*x) satisfies the differential equation (1 - 2*x)*y' = (3 - 2*x)*y. R. J. Mathar's recurrence above follows easily from this.
The sequence b(n) := 2^n*n! also satisfies R. J. Mathar's recurrence with b(0) = 1 and b(1) = 2. This leads to the continued fraction representation a(n) = 2^n*n!*( 1 + 1/(2 - 2/(5 - 4/(7 - ... - (2*n - 2)/(2*n + 1) )))) for n >= 2. Taking the limit gives the continued fraction representation exp(1/2) = 1 + 1/(2 - 2/(5 - 4/(7 - ... - (2*n - 2)/((2*n + 1) - ... )))). (End)
a(n) = 2^n*KummerU(-n, -n, 1/2). - Peter Luschny, May 10 2022
a(n) = n!*2^n*hypergeom([-n], [-n], 1/2). - Peter Luschny, Jul 28 2024
EXAMPLE
a(3) = 2*3*a(2) + 1 = 6*13 + 1 = 79.
G.f. = 1 + 3*x + 13*x^2 + 79*x^3 + 633*x^4 + 6331*x^5 + 75973*x^6 + 1063623*x^7 + ...
MAPLE
G:=(x, a, k, n)-> n!*a^n*x + k*sum(n!*a^(n-j)/j!, j=1..n); seq(G(1, 2, 1, n), n = 0..20) # Gary Detlefs, Mar 26 2018
a := n -> 2^n*add((n!/k!)*(1/2)^k, k=0..n):
seq(a(n), n=0..19); # Peter Luschny, Jan 06 2020
seq(simplify(2^n*KummerU(-n, -n, 1/2)), n = 0..19); # Peter Luschny, May 10 2022
MATHEMATICA
Table[ Gamma[ n, 1/2 ]*Exp[ 1/2 ]*2^(n-1), {n, 1, 24} ]
...and/or... s=1; lst={}; Do[s+=s++n; AppendTo[lst, s], {n, 1, 5!, 2}]; lst (* Vladimir Joseph Stephan Orlovsky, Oct 23 2008 *)
a[ n_] := If[ n<0, 0, Floor[ n! E^(1/2) 2^n ]] (* Michael Somos, Sep 04 2013 *)
nxt[{n_, a_}]:={n+1, 2*a(n+1)+1}; NestList[nxt, {0, 1}, 20][[All, 2]] (* Harvey P. Dale, Jan 06 2022 *)
a[n_] := n! 2^n Hypergeometric1F1[-n, -n, 1/2];
Table[a[n], {n, 0, 19}] (* Peter Luschny, Jul 28 2024 *)
PROG
(PARI) {a(n) = if( n<0, 0, n! * sum(k=0, n, 2^(n-k) / k!))} /* Michael Somos, Sep 04 2013 */
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
EXTENSIONS
Better description and formulas from Michael Somos
STATUS
approved